Question

Use Descartes's Rule of Signs to determine the possible numbers of positive and negative real zeros of f(x) = 4x^4 - 2x³-x²-2x+8.What is the possible number of positive real zeros?(Use a comma to separate answers as needed.)What is the possible number of negative real zeros?(Use a comma to separate answers as needed.)

Answer 1

• Possible number of positive real zeros: 2, 0

• Possible number of negative real zeros: 2, 0

Step-by-step explanation

• Descartes's Rule of Signs: ,criterion based on the number of sign changes in the sequence of coefficients of the polynomial. It gives an upper bound on the number of positive or negative real roots of a polynomial with real coefficients.

Thus, based on this criterion, we can count the number of sign changes in our polynomial:

The number of positive roots of f(x) is either equal to the number of sign changes in f(x) or less than the number of sign changes by an even number.

Meaning that in this case, we have 2 or 0 positive roots.

The same rule applies to find the number of negative real zeros as well, but we count the sign changes of f(-x):

Also means that in this case, we have 2 or 0 positive roots.